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4.9. Other Problems 155

4.9 Other Problems

1. For which complex numbers a is the mapping

one-one on the closed unit disc (1~1 5 l)?

2. The mapping z - z2 maps a straight line in the complex plane onto
   a parabola. Identify the vertex of the parabola.


3. Find necessary and sufficient conditions on p, q, r that the roots of

x3+px2+qx+r=0

are the vertices of an equilateral triangle in the complex plane.

4. Find, in terms of a, b, c, a formula for the area of a triangle in the
   complex plane whose vertices are the roots of

x3 - ax2 + bx - c = 0.

5. Show that a necessary and sufficient condition that a real cubic equa-
   tion ax3 + bx2 + cx + d = 0 have one real and two pure imaginary
   roots is that bc = ad and ac > 0.


6. Let a, b, c, d be complex numbers, all with absolute value equal to
   unity. Prove that, in the unit circle with center 0, the polynomial
   az3 + bz2 + cz + d has a maximum absolute value not less than &.


7. Let p(x) be a polynomial of positive degree n and let 0 < m 5 n.
   Suppose that co, cl, cz,... , c, are constants for which

co + Cl2 + c2x2 +. *. + C,~,~lxn-m-l + cn-+p(x)

+ c,-,+lp(x + 1) +... + C,P(X + m) = 0

identically. Show that CO = cl =... = c,, = 0.

8. f(t) is a polynomial of degree n over C such that a power of f(t)
   is divisible by a power of f’(t), i.e. [f(t)]” is divisible by [f’(t)]9 for
   some positive integers p and q.
   Prove that f(t) is divisible by f’(t) and that f(t) has a single zero of
   multiplicity n.


9. The nonconstant polynomials p(t) and q(t) over C have the same set
   of numbers for their zeros, but with possibly different multiplicities.
   The same is true of the polynomials p(t) + 1 and q(t) + 1. Prove that
   p(t) = q(t).